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Important!

Introductory papers

• Quantum Geometry and New Concept of Space
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• Introduction to Quantum Principal Bundles
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• Physics Beyond Limits of Quantum Uncertainty Relations
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•  Diagrammatic Formulation of Multi-braided Quantum Groups
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My papers on Quantum Principal Bundles

Quantum Principal Bundles: Proceedings of XXIIth International Conference on Differential-Geometric Methods in Theoretical Physics, Ixtapa-Zijuatanejo, Mexico, September 1993

Abstract. This is a very brief exposition of my theory of quantum principal bundles. The exposition does not contain the proofs, however it is relatively self-contained.

Geometry of Quantum Principal Bundles I: Commun Math Phys 175 (3) 457-521 (1996)

Abstract. This work contains a detailed exposition of the theory of quantum principal bundles, in the context of locally-trivial structures over classical smooth manifolds. This version of the theory already contains the basic structural elements of the fully quantum theory of principal bundles. The classification problem of such locally-trivial quantum principal bundles is resolved, and an intrinsic differential calculus is constructed. The formalism of connections is developed, including the analogs of all the basic entities of the classical theory. Local expressions for connections, curvatures, covariant derivatives and (pseudo)tensorial forms are obtained. Various interesting and surprising purely quantum phenomenas appear in the theory, and they are extensively illustrated in the example of bundles with the quantum SU(2) structure group.

Geometry of Quantum Principal Bundles II: Rev Math Phys 9 (5) 531-607 (1997)

Abstract. In this work I developed a general theory of quantum principal bundles, incorporating the formalism of the previous paper into the fully quantum context where the base manifold, the structure group and the bundle are all considered as quantum objects. A particular attention is given to differential calculus on quantum principal bundles, and to the formalism of connections--including the constructions of horizontal projections, covariant derivative operators and the curvature tensors. Among other topics covered in the paper we find infinitezimal gauge transformations, a construction of a quantum Weil homomorphism, and various constructive approaches to differential calculus on quantum principal bundles. Interesting examples are considered, too.

General Spinor Structures on Quantum Spaces: Preprint, Institute of Mathematics, UNAM

Abstract. A general theory of quantum spinor structures on quantum spaces is presented, within the conceptual framework of the formalism of quantum principal bundles. Quantum analogs of all basic objects of the classical theory are constructed and analyzed. This includes Laplace and Dirac operators, a Hodge *-operator, and their mutual relations, as well as quantum versions of Clifford and spinor bundles. Furthermore, various integration operators are defined. Interesting examples and constructions are included. We also present a self-contained formalism of braided Clifford algebras. A special attention is given to the study of purely quantum phenomena appearing in the theory.

Quantum Principal Bundles and Corresponding Gauge Theories: J Phys A Math Gen 30 2027-2054 (1997)

Abstract. A generalization of classical gauge theory is presented, in which compact quantum groups play the role of the internal symmetry groups. All considerations are performed in the framework of my noncommutative-geometric formalism of quantum principal bundles. Quantum counterparts of classical gauge bundles and classical gauge transformations are introduced and investigated. A natural differential calculus on quantum gauge bundles is constructed and analyzed. Kinematical and dynamical properties of corresponding gauge theories are discussed. Particular attention is given to the purely quantum phenomenas appearing in the formalism, and their physical interpertation. An example with quantum SU(2) group is considered.

Quantum Principal Bundles and Tannaka-Krein Duality Theory: Rep Math Phys 38 (3) 313-324 (1996)

Abstract. The structure of quantum principal bundles is studied, from the viewpoint of Tannaka-Krein duality theory. It is shown that if the structure quantum group is compact, principal bundles over a given quantum space are in a natural correspondence with certain contravariant functors defined on the category of finite-dimensional unitary representations of the structure group, with the values in the category of finite projective bimodules over a *-algebra representing the base space. This establishes a connection between quantum principal bundles and associated vector bundles.

Differential Structures on Quantum Principal Bundles: Rep Math Phys 41 (1) 91-115 (1998)

Abstract. A fully constructive approach to differential calculus on quantum principal bundles is presented. The calculus on the bundle is built in an intrinsic way, starting from given graded (differential) *-algebras representing differential forms on the bundle and differential forms on the quantum base space, together with the family of antiderivations acting on horizontal forms, used as counterparts of local trivialisations. In this conceptual framework, a natural differential calculus on the structure quantum group is described. Higher-order calculi on the structure quantum group coming from both universal envelopes and braided exterior algebras are considered.

Quantum Classifying Spaces and Universal Quantum Characteristic Classes: Banach Center Publications, 40 315-327 (1997)

Abstract. A construction of the noncommutative-geometric counterparts of classical classifying spaces is presented, for general compact matrix quantum structure groups. A quantum analogue of the classical concept of the classifying map os introduced and analyzed. Interrelations with the abstract algebraic theory of quantum characteristic are discussed. Various non-equivalent approaches to defining universal characteristic classes are outlined.

Quantum Principal Bundles and Their Characteristic Classes: Banach Center Publications, 40 303-313 (1997)

Abstract. A general theory of characteristic classes of quantum principal bundles is sketched, incorporating basic ideas of classical Weil theory into the conceptual framework of non-commutative differential geometry. A purely cohomological interpretation of the Weil homomorphism is given, together with a geometrical interpretation via quantum invariant polynomials.

Characteristic Classes of Quantum Principal Bundles: Preprint, Institute of Mathematics, UNAM

Abstract: This work is devoted to a detailed presentation of the theory of quantum characteristic classes, in the conceptual framework of quantum principal bundles. In particular, a noncommutative-geometric generalization of classical Weil construction is presented. A special attention is given to the special case when the bundle does not admit regular connections. A cohomological description of the domain of the Weil homomorphism is given. Relations between universal characteristic classes for the regular and the general case are analyzed. In analogy with classical geometry, a natural spectral sequence is introduced and investigated. The appropriate counterpart of the Chern character is constructed, for structures admitting regular connections. Various illustrative examples and constructions are presented.

General Frame Structures on Quantum Principal Bundles: Rep Math Phys (to appear)

Abstract. A quantum generalization of the classical formalism of frame bundles is developed. This is done within a general conceptual framework of the theory of quantum principal bundles, incorporating into the theory the concept of a Levi-Civita connection. The theory includes classical Riemanian geometry and symplectic geometry as special cases. The construction of a natural differential calculus on quantum principal frame bundles is presented, including the construction of the associated differential calculus over the quantum structure group. Explicit expressions allowing the calculations of all the basic entities of the formalism are derived. Various interesting examples are considered. General quantum torsion operators are defined and their properties are studied.

Affine Structures on Quantum Principal Bundles: Preprint, Institute of Mathematics, UNAM

Abstract. Quantum affine bundles are quantum principal bundles with affine quantum structure groups. A general theory of quantum affine bundles is presented. A detailed analysis of differential calculi over these bundles is performed, including the construction of a natural differential calculus over the structure affine quantum group. A special attention is given to the study of the specific properties of quantum affine connections, and varios purely quantum phenomenas appearing in the context of quantum affine bundles. Interesting constructions are presented. In particular, the main ideas are illustrated within the example of the quantum Hopf fibration.

Classical Spinor Structures on Quantum Spaces: Clifford Algebras and Spinor Structures, Kluwer, 365-377 (1995)

Abstract. A noncommutative-geometric generalization of the classical concept of spinor structure is presented. This is done in the framework of the formalism of quantum principal frame bundles. The base space and the bundle are considered as quantum objects, however the structure group is a classical Spin group. Constructions of the corresponding Dirac operator and the Laplacian are presented, and their properties are studied. Examples of quantum spin manifolds are discussed.

On Framed Quantum Principal Bundles: Preprint, Institute of Mathematics, UNAM

Abstract. The paper deals with quantum frame bundles possessing classical structure groups (but quantum base manifolds). A general theory of quantum frame bundles simplifies in this special context, however it is still possible to illustrate various purely quantum phenomenas. Such structures have a potential interest in the formulation of general relativity theory over a quantum space-time, because the idea of quantum fluctuations at the Planck scale (characteristic of a quantum space-time) is logically independent of the assumpions about the nature of local symmetries.

Quantum Principal Bundles and Hopf-Galois Extensions: Preprint, Institute of Mathematics, UNAM

Abstract. Using the representation theory of compact quantum groups, it is shown that every quantum principal bundle with a compact structure group is a Hopf-Galois extension. Furthermore, it is shown that every differential calculus over such a quantum principal bundle is a graded-differential version of the Hopf-Galois extension. Various interesting algebraic identities are derived.

Quantum Gauge Transformations and Braided Structure on Quantum Principal Bundles: Preprint, Institute of Mathematics, UNAM

Abstract. It is shown that every quantum principal bundle is intrinsically braided, in the sense that there exists a natural braid operator twisting the elements of the *-algebra representing (smooth functions on) the bundle. In classical geometry, this braid operator reduces to the standard transposition. Algebraic properties of this braid operator are studied in detail, including its natural extension to the level of differential forms on the bundle. Furthermore, quantum gauge bundles are introduced and investigated, with the help of the mentioned braided structure. In particular, a natural action of the quantum gauge bundle on the initial principal bundle is constructed. This can be viewed as a generalization of classical gauge transformations. Transformation properties of various fundamental objects appearing in the formalism of quantum principal bundles are studied.

Papers on Quantum & Braided Structures, and Their Applications

Generalized Braided Quantum Groups: Isr Jour Math 98 329-348 (1997)

Abstract. A generalization of the theory of Hopf algebras is presented. The generalization overcomes an inherent geometrical inhomogeneity of standard quantum groups and braided quantum groups, in the sense of allowing completely "pointless" objects. All braid-type equations appear as a consequence of deeper axioms. Braided counterparts of basic algebraic relations between fundamental entities of the standard theory are found.

Clifford Algebras and Spinors in Braided Geometry: Adv Appl Cliff Alg (Proc Suppl) 4 (s1) (1994)--This is a joint work with Z Oziewicz

Abstract. The paper provides an introduction to Clifford algebras and spinors for an arbitrary braid. Braided Clifford algebras are defined as Chevalley-Kahler deformations of braided exterior algebras (the Woronowicz algebras). Spinor representations are introduced, following classical Cartan's approach.

Braided Clifford Algebras as Braided Quantum Groups: Adv Appl Cliff Alg 4 (2) 145-156 (1994)

Abstract. The paper deals with braided Clifford algebras, understood as Chevalley-Kahler deformations of braided exterior algebras. It is shown that such Clifford algebras can be naturally endowed with a braided quantum group structure. Basic group entities are constructed explicitly.

Quantum Clifford Algebras from Spinor Representations: Jour Math Phys 37 (11) 5747-5775 (1996)--This is a joint work with R Bautista, A Criscuolo, M Rosenbaum and J D Vergara

Abstract. A general theory of quantum Clifford algebras is presented, based on a quantum generalization of the Cartan theory of spinors. We concentrate on the case when it is possible to apply the quantum-group formalism of bicovariant bimodules, assuming that a splitting of the basic vector space into the sum of two isotropic subspaces is fixed. This allows us to interpret the quadratic form as a natural braided-pairing between the two isotropic subspaces. The corresponding spinor representations are investigated. Starting from our Clifford algebras we introduce the quantum-Eucledean underlying spaces from where the analogues of Dirac and Laplace operators are built.

First-Order Differential Calculi Over Multi-Braided Quantum Groups: Preprint, Institute of Mathematics, UNAM/Belgrade University, Serbia

Abstract. A differential calculus of the first order over multi-braided quantum groups is developed. In analogy with the standard theory, left/right-covariant and bicovariant differential structures are introduced and investigated. Furthermore, antipodally covariant calculi are studied. The concept of the *-structure on a multi-braided quantum group is formulated, and in particular the structure of left-covariant *-covariant calculi is analyzed. A special attention is given to differential calculi covariant with respect to the action of the associated braid system. In particular it is shown that the left/right braided-covariance appears as a consequence of the left/right-covariance relative to the group action. Braided counterparts of all basic results of the standard theory are found.

Higher-order Measures, Generalized Quantum Mechanics and Hopf Algebras: Mod Phys Lett A, Vol 19, No 3, 197-212 (2004)--This is a joint work with Chryssomalis Chryssomalakos

Abstract. In this puzzling paper, Chryssomalis and Micho study Sorkin's proposals of a generalization of quantum mechanics. They find that the theories proposed derive their probabilities from k-th order polynomials in additive measures, in the same way that quantum mechanics uses a probability bilinear in the quantum amplitude and its complex conjugate. Two complementary approaches are presented, a C* and a Hopf-algebraic one, illuminating both algebraic and geometric aspects of the problem.

Assembly Language Programming & Related Works/2

Presentation Manager Programming In Assembly Language: Preprint, Q-Systems & Institute of Mathematics, UNAM

Abstract. This is the first of a series of articles in which we study programming schemes for OS/2 operating system, based on assembly language. Here we focus on basic Presentation Manager programming. Principal techniques are illustrated within an interesting sample program involving random creation of rectangles. In this conceptual framework, we explain the algebraic formalism behind multiply-with-carry random number generators.

OS/2 WorkPlace Shell Programming In Assembly Language: Preprint, Q-Systems & Institute of Mathematics, UNAM

Abstract. We present basic ideas and techniques of constructing OS/2 libraries for WorkPlace Shell objects in assembly language. In such a way we overcome various inherent obstacles of the standard approach, and achieve a complete control over the generated binary code. A particular attention is given to method overrides, new object methods, and important SOM kernel functions with associated data structures. Illustrative examples are presented.

Installing BSD Operating Systems on IBM Netvista S40: Preprint, Q-Systems & Institute of Mathematics, UNAM

Abstract. We present several ways of setting up FreeBSD operating system on IBM Netvista S40, a so-called "legacy free" computer. The difficulty arises because the machine has no standard AT keyboard controller, and the existing FreeBSD subroutines for the gate A20 and for the keyboard controller probes result inappropriate. We discuss a replacement bootstrap code, which more carefully deals with the A20 issue. Some simple modifications to the FreeBSD kernel code are considered, too. A manual method for preparing a bootable installation CD, suitable for both Netvista and all standard configurations, is examined. Installations of DragonFly, NetBSD, OpenBSD and OS/2 are also discussed.

Open Object Rexx for DragonFly/FreeBSD: Binary Package & Source Code

Description. This is my DragonFly/FreeBSD port of Open Object Rexx, a powerful object-oriented scripting language. The interpreter is almost fully compatible with the original Object Rexx by IBM. The port is derived from the generic UNIX source code, with the idea to add BSD-specific enhancements over time and also to develop modules that would allow to run the most of OS/2 Object Rexx code on BSD platforms. Assembly language interface to Open Object Rexx will be discussed in a forthcoming paper.

Quetzal-A Live DVD/CD System Based on OpenBSD

Abstract. This is the companion article to Quetzal-which is a live DVD/CD based on OpenBSD operating system. We explain basic features of the system, a DHCP/PXE remote-booting setup, and provide more detailed instructions for a manual hard disk install. We also discuss simple modifications to the OpenBSD kernel used in Quetzal, and some tricks, that might be interesting on their own.